Lebesgue approximation of $(2,\beta)$-superprocesses

TL;DR

This paper demonstrates that for certain superprocesses in Euclidean space, the random measure at fixed time can almost surely be approximated by Lebesgue measure on neighborhoods of its support, extending previous results.

Contribution

It extends Lebesgue approximation results from Dawson-Watanabe superprocesses to $(2,eta)$-superprocesses with $eta<1$ in higher dimensions.

Findings

Almost sure Lebesgue approximation of $\xi_t$ by neighborhoods of its support.

Extension of Lebesgue approximation to $(2,eta)$-superprocesses.

Use of truncation and hitting probability bounds in proofs.

Abstract

Let $\xi=(\xi_t)$ be a locally finite $(2,\beta)$-superprocess in $\RR^d$ with $\beta<1$ and $d>2/\beta$. Then for any fixed $t>0$, the random measure $\xi_t$ can be a.s. approximated by suitably normalized restrictions of Lebesgue measure to the $\varepsilon$-neighborhoods of ${\rm supp}\,\xi_t$. This extends the Lebesgue approximation of Dawson-Watanabe superprocesses. Our proof is based on a truncation of $(\alpha,\beta)$-superprocesses and uses bounds and asymptotics of hitting probabilities.